If you remove the API keys for Google, LinkedIn and Twitter, then the login buttons for those social networks won't display and a user won't be able to register or login with them. By leaving just the Facebook API key set, that will be the only one used for authentication. You'll still be able to use the social buttons and many of the widgets from the other networks though.
I hope that helps answer your question, but if you were asking something else, just let me know. Also, if you want us to investigate the issues you're having with the other 3, just let us know... we'll gladly help however we can.
Thanks,
Alex